Phi, Φ, 1.618…, has two properties that make it unique among all numbers.
- If you square Phi, you get a number exactly 1 greater than itself: 2.618…, or
Φ² = Φ + 1.
- If you divide Phi into 1 to get its reciprocal, you get a number exactly 1 less than itself: 0.618…, or
1 / Φ = Φ – 1.
These relationships are derived from the dividing a line at its golden section point, the point at which the ratio of the line (A) to the larger section (B) is the same as the ratio of the larger section (B) to the smaller section (C).
This relationship is expressed mathematically as:
A = B + C, and
A / B = B / C.
Solving for A, which on both sides give us this:
B + C = B²/C
Let’s say that C is 1 so we can determine the relative dimensions of the line segments. Now we simply have this:
B + 1 = B²
This can be rearranged as:
B² – B – 1 = 0
Note the various ways that this equation can be rearranged to express the relationship of the line segments, and also Phi’s unique properties:
B2 = B + 1
1 / B = B – 1
B2 – B1 – B0 = 0
Note: Bx means n raised to the x power. Some browsers may not display exponents as superscripts or raised characters.
Now we have a formula that can be solved using the Quadratic formula. This formula allows you to solve a quadratic equation for an unknown x, with a, b, and c as constants. A quadratic equation has this form:
-
ax² + bx + c = 0
The solution to this is found with the quadratic formula:
So our formula for the golden ratio above (B2 – B1 – B0 = 0) can be expressed as this:
1a2 – 1b1 – 1c = 0
The solution to this equation using the quadratic formula is (1 plus or minus the square root of 5) divided by 2:
( 1 + √5 ) / 2 = 1.6180339… = Φ
( 1 – √5 ) / 2 = -0.6180339… = -Φ
The reciprocal of Phi (denoted with an upper case P), is known often as by phi (spelled with a lower case p).
Phi, curiously, can also be expressed all in fives as:
5 ^ .5 * .5 + .5 = Φ
This provides a great, simple way to compute phi on a calculator or spreadsheet!
Here’s a little more phi mathemagic, contributed by Abe Ihmeari:
Φ * √5 = 3.6180339… = Φ + 2
Determining the nth number of the Fibonacci series
You can use phi to compute the nth number in the Fibonacci series (fn):
fn = Φ n / 5½
As an example, the 40th number in the Fibonacci series is 102,334,155, which can be computed as:
f40 = Φ 40 / 5½ = 102,334,155
This method actually provides an estimate which always rounds to the correct Fibonacci number.
You can compute any number of the Fibonacci series (fn) exactly with a little more work:
fn = [ Φ n – (1-Φ)n ] / √5
Note: √5 can be expressed as 2Φ-1 to use Φ for all the terms above.
Determining Phi with Trigonometry and Limits
Phi can be related to e, the base of natural logs,
through the inverse hyperbolic sine function:
Φ = e ^ asinh(.5)
It can be expressed as a limit:
or
Other unusual phi relationships
There are many unusual relationships in the Fibonacci series. For example, for any three numbers in the series Φ(n-1), Φ(n) and Φ(n+1), the following relationship exists:
Φ(n-1) * Φ(n+1) = Φ(n)2 – (-1)n
( e.g., 3*8 = 52-1 or 5*13=82+1 )
Here’s another:
Every nth Fibonacci number is a multiple of Phi(n),
where Phi(n) is the nth number of the Fibonacci sequence.
Given 0, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946
(Every 3rd number., e.g., 2, 8, 34, 144, is a multiple of Phi(3), which is 2)
(Every 4th number, e.g., 3, 21, 144 and 987, is a multiple of Phi(4), which is 3)
(Every 5th number, e.g., 5, 55, 610, and 6765, is a multiple of Phi(5), which is 5)
(Every 6th number, e.g., 8, 144, 2584, is a multiple of Phi(6), which is 8)
(Every 7th number, e.g., 13, 377, 10946, is a multiple of Phi(7), which is 13)
And, as contributed by Abe Ihmeari, for any Fibonacci sequence number f(n), we find that f(n)-f(n-5)-f(n-10) = 10 ∙ f(n-5). This is easiest to see when the Fibonacci sequence numbers are grouped in fives. As an example, 4181-377-34 = 3770, which is 10 ∙ 377, and 28657-2584-233 = 25840, which is 10 ∙ 2584!
| 0 | 5 | 55 | 610 | 6765 |
| 1 | 8 | 89 | 987 | 10946 |
| 1 | 13 | 144 | 1597 | 17711 |
| 2 | 21 | 233 | 2584 | 28657 |
| 3 | 34 | 377 | 4181 | 46368 |
And another:
The first perfect square in the Fibonacci series, 144,
is number 12 in the series and its square root is 12!
0, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144
or, if not starting with 0:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144